## Random Geometric Graphs

*Mathew Penrose*

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198506263
- eISBN:
- 9780191707858
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198506263.001.0001
- Subject:
- Mathematics, Probability / Statistics

This book sets out a body of rigorous mathematical theory for finite graphs with nodes placed randomly in Euclidean d-space according to a common probability density, and edges added to connect ... More

## New Perspectives in Stochastic Geometry

*Wilfrid S. Kendall and Ilya Molchanov (eds)*

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199232574
- eISBN:
- 9780191716393
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.001.0001
- Subject:
- Mathematics, Geometry / Topology

Stochastic geometry is a subject with roots stretching back at least 300 years, but one which has only been formed as an academic area in the last 50 years. It covers the study of random patterns, ... More

## Very Fast Oscillations

*Roger D. Roger and Miles A. Whittington*

### in Cortical Oscillations in Health and Disease

- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780195342796
- eISBN:
- 9780199776276
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195342796.003.0010
- Subject:
- Neuroscience, Molecular and Cellular Systems, Development

VFO occurs in in vitro models when chemical receptors are blocked. In particular, VFO does not require GABAA receptors, even though interneurons fire at high rates during in vivo very fast ... More

## The Pleasures and Pains of Studying the Two-Type Richardson Model

*Maria Deijfen and Olle Häggström*

### in Analysis and Stochastics of Growth Processes and Interface Models

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199239252
- eISBN:
- 9780191716911
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199239252.003.0002
- Subject:
- Mathematics, Probability / Statistics, Analysis

This chapter provides a survey of known results and open problems for the two-type Richardson model, which is a stochastic model for competition on a lattice. In its simplest formulation, the ... More

## Applications of the Lace Expansion to Statistical-Mechanical Models

*Akira Sakai*

### in Analysis and Stochastics of Growth Processes and Interface Models

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199239252
- eISBN:
- 9780191716911
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199239252.003.0006
- Subject:
- Mathematics, Probability / Statistics, Analysis

Synergetics is a common feature in interesting statistical-mechanical problems. One of the most important examples of synergetics is the emergence of a second-order phase transition and critical ... More

## Percolation and epidemics

*Sergey N. Dorogovtsev*

### in Lectures on Complex Networks

- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780199548927
- eISBN:
- 9780191720574
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199548927.003.0006
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter considers the organization of connected components in uncorrelated networks, in particular, the structure and size of a giant connected component. These properties are closely related to ... More

## Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

*Geoffrey Grimmett and Colin McDiarmid (eds)*

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780198571278
- eISBN:
- 9780191718885
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198571278.001.0001
- Subject:
- Mathematics, Probability / Statistics

Professor Dominic Welsh has made significant contributions to the fields of combinatorics and discrete probability, including matroids, complexity, and percolation. He has taught, influenced, and ... More

## PERCOLATIVE INGREDIENTS

*Mathew Penrose*

### in Random Geometric Graphs

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198506263
- eISBN:
- 9780191707858
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198506263.003.0009
- Subject:
- Mathematics, Probability / Statistics

This chapter contains some known results on connectivity which are used later on. The notion of unicoherence of a simply-connected set is explained and extended to lattices. Peierls (counting) ... More

## Stochastic Geometry and Telecommunications Networks

*Sergei Zuyev*

### in New Perspectives in Stochastic Geometry

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199232574
- eISBN:
- 9780191716393
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.003.0016
- Subject:
- Mathematics, Geometry / Topology

Just as queueing theory revolutionized the study of circuit switched telephony in the twentieth century, stochastic geometry is gradually becoming a necessary theoretical tool for modelling and ... More

## Percolation and Random Graphs

*Remco van der Hofstad*

### in New Perspectives in Stochastic Geometry

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199232574
- eISBN:
- 9780191716393
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.003.0006
- Subject:
- Mathematics, Geometry / Topology

In this chapter, we define percolation and random graph models, and survey the features of these models.

## Fractured Porous Media

*Pierre M. Adler, Jean-François Thovert, and Valeri V. Mourzenko*

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199666515
- eISBN:
- 9780191748639
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199666515.001.0001
- Subject:
- Physics, Condensed Matter Physics / Materials

This book aims to estimate the macroscopic properties of fractures, fracture networks and fractured porous media from easily measurable quantities. Attention is focused on geological media where ... More

## Applications to Other Wh-Fronting Languages, Pied-Piping, and Intervention Effects

*Seth Cable*

### in The Grammar of Q: Q-Particles, Wh-Movement, and Pied-Piping

- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780195392265
- eISBN:
- 9780199866526
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195392265.003.0004
- Subject:
- Linguistics, Syntax and Morphology

This chapter begins the argument that the ‘Q-based’ account should be extended to all other wh-fronting languages. It first presents some general typological and learning-theoretic arguments. These ... More

## Optimization

*Sergey N. Dorogovtsev*

### in Lectures on Complex Networks

- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780199548927
- eISBN:
- 9780191720574
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199548927.003.0014
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter demonstrates how a universal requirement for optimality leads to the complex structural organization of a network. It discusses a long-lasting criticism of the preferential concept and ... More

## HOPPING CONDUCTIVITY

*V.F. Gantmakher*

### in Electrons and Disorder in Solids

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198567561
- eISBN:
- 9780191718267
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567561.003.0004
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter is dedicated to electron transitions between localized states. After a description of the states with the help of localization radius and the general expression for the transition ... More

## Percolation and network resilience

*Mark Newman*

### in Networks

- Published in print:
- 2018
- Published Online:
- October 2018
- ISBN:
- 9780198805090
- eISBN:
- 9780191843235
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198805090.003.0015
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

A discussion of the site percolation process on networks and its application as a model of network resilience. The chapter starts with a description of the percolation process, in which nodes are ... More

## ELECTRON MOBILITY IN DENSE HE GAS

*A.F. Borghesani*

### in Ions and electrons in liquid helium

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780199213603
- eISBN:
- 9780191707421
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213603.003.0027
- Subject:
- Physics, Condensed Matter Physics / Materials

Experiments on the mobility of electrons in dense helium gas elucidated how localized electron states develop when the gas density gas is increased. Up to 77 K, the density dependence of the mobility ... More

## First Steps in Random Walks: From Tools to Applications

*J. Klafter and I. M. Sokolov*

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199234868
- eISBN:
- 9780191775024
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199234868.001.0001
- Subject:
- Physics, Soft Matter / Biological Physics

The name “random walk” for a problem of a displacement of a point in a sequence of independent random steps was coined by Karl Pearson in 1905 in a question posed to readers of “Nature”. The same ... More

## Percolation and network resilience: A discussion of one of the simplest of processes taking place on networks, percolation, and its use as a model of network resilience

*M. E. J. Newman*

### in Networks: An Introduction

- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780199206650
- eISBN:
- 9780191594175
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199206650.003.0016
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter examines one of the simplest network processes, which is percolation. This leads to an elegant theory concerning the robustness of networked systems to the failure of their components. ... More

## Random systems

*Hidetoshi Nishimori and Gerardo Ortiz*

### in Elements of Phase Transitions and Critical Phenomena

- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199577224
- eISBN:
- 9780191722943
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199577224.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Real materials always contain randomness or disorder that cannot be expressed by idealized simple model systems. The present chapter studies the effects of randomness on phase transitions and ... More

## Classical Percolation, Generalized Percolation and Cascades

*Ginestra Bianconi*

### in Multilayer Networks: Structure and Function

- Published in print:
- 2018
- Published Online:
- July 2018
- ISBN:
- 9780198753919
- eISBN:
- 9780191815676
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198753919.003.0012
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter characterizes the robustness of multiplex and multilayer networks using classical percolation, directed percolation and antagonistic percolation. Classical percolation determines whether ... More